GAMs and GPs

probability
generalised additive models
gaussian processes
Author

Nathaniel Forde

Published

February 15, 2024

import pandas as pd
import matplotlib.pyplot as plt
import bambi as bmb
import seaborn as sns
from pygam.datasets import mcycle
from pygam import LinearGAM, s, f, GAM, l, utils
import numpy as np
import arviz as az
import pymc as pm


random_seed = 100

import warnings

warnings.simplefilter('ignore')

Flexibility and Calibration

In this blog post we’re going to dive into modelling of non-linear functions and explore some of the tooling available in the python eco-system. We’ll start by looking into Generalised Additive Models with splines in pyGAM before preceding to look Bayesian versions of spline modelling comparing the splines to Gaussian processes in Bambi and PyMC

Our interest in these models stems from their flexibility to approximate functions of arbitrary complexity. We’ll see how the methods work in the case of relatively straightforward toy example and then we’ll apply each of the methods to deriving insights into the functional form of insurance loss curves. In this application we adapt a data set discussed in Mick Cooney’s Stan case study to demonstrate the power of hierarchical spline models. Throughout we’ll draw on the discussion of these methods in Osvaldo Martin’s “Bayesian Analysis with Python” for practical details implementing these models.

All of these methods need to be assessed with respect to their in-sample model fit and their out of sample performance. How can we best calibrate the model fits to perform reasonably well out of sample?

Generalised Additive models

The canonical reference for GAMs is Simon Wood’s “Generalised Additive Models: An Introduction with R” which outlines in some detail the theoretical background of splines and univariate smoothers. The book stresses the trade-offs between the flexibility of splines and the need for cross-validation and penalised estimation methods for spline based modelling.

In R these penalised models fits can be achieved in mgcv which incorporates a wilkinson like formula syntax for model specification: y ~ s(x) + s(x1). The closest implementation in python is available in PyGam and we will adopt this package to illustrate the application of smoothing terms and the penalities.

PyGAM and Penalised Fits

In the next code block we load in an example data set on which to demonstrate univariate smoothing patterns using penalised splines. These models can be optimised by fitting differing strength penalities over a varying number of splines

X, y = mcycle(return_X_y=True)

fig, ax = plt.subplots(figsize=(10, 6))
ax.scatter(X, y)
ax.set_ylabel("Acceleration")
ax.set_xlabel("Time Step")
ax.set_title("Crash Test Dummy Acceleration \n Simulated Motorcycle Crash", fontsize=20)
Text(0.5, 1.0, 'Crash Test Dummy Acceleration \n Simulated Motorcycle Crash')

Next we fit a number of different models to account for the herky-jerky of the data generating processs. We vary the parameterisations to see how the numbers of splines and strength of the penalty help account for the variation in \(y\) over the support of \(X\).

gam1 = LinearGAM(s(0, n_splines=5)).fit(X, y)
gam2 = LinearGAM(s(0, n_splines=7)).fit(X, y)
gam3 = LinearGAM(s(0, n_splines=10)).fit(X, y)
gam4 = LinearGAM(s(0, n_splines=15)).fit(X, y)
gam5 = LinearGAM(s(0, lam=.1)).fit(X, y)
gam6 = LinearGAM(s(0, lam=.5)).fit(X, y)
gam7 = LinearGAM(s(0, lam=5)).fit(X, y)
gam8 = LinearGAM(s(0, lam=15)).fit(X, y)


def plot_fit(gam, X, y, ax, t, c1='b', c2='r'):
    XX = gam.generate_X_grid(term=0, n=500)

    ax.plot(XX, gam.predict(XX), color=c2, linestyle='--')
    ax.plot(XX, gam.prediction_intervals(XX, width=.95), color=c1, ls='--')

    ax.scatter(X, y, facecolor='gray', edgecolors='none')
    ax.set_title(f"""95% prediction interval with {t} \n LL: {gam.statistics_['loglikelihood']}""");

fig, axs = plt.subplots(4,2, figsize=(10, 20))
axs = axs.flatten()
titles = ['5_splines', '7_splines', '10_splines', '15_splines',
'lam=.1', 'lam=.5', 'lam=5', 'lam=15']
gs = [gam1, gam2, gam3, gam4, gam5, gam6, gam7, gam8]
for ax, g, t in zip(axs, gs, titles):
    plot_fit(g, X, y, ax, t)

Here we’ve seen the PyGAM package applied to fitting our model to the data. In the formula specification we see y ~ s(i) where i denotes the index of the column variable in the X data.

Over the range of of the x-axis we can see how the conditional expectation is more or less well fit to the data depending on how the penalities and complexity of the model is specified.

Optimising The Parameter Setting

We can see from the model summary what is going on under the hood. For a given model specification the summary will report a number of model-fit statistics such as the log-likelihood and the AIC.

## Naive Model manually specified splines
gam_raw = LinearGAM(s(0)).fit(X, y)
print("log_likelihood:", gam_raw.statistics_['loglikelihood'])
print("AIC:", gam_raw.statistics_['AIC'])
log_likelihood: -954.5558010528869
AIC: 1931.5965064139743

The question then becomes what changes are induced in the model as we seek to optimise these model fit statistics.

## model optimised
gam = LinearGAM(s(0),  fit_intercept=False)
gam.gridsearch(X, y)
gam.summary()
  0% (0 of 11) |                         | Elapsed Time: 0:00:00 ETA:  --:--:--
  9% (1 of 11) |##                       | Elapsed Time: 0:00:00 ETA:   0:00:00
100% (11 of 11) |########################| Elapsed Time: 0:00:00 Time:  0:00:00
LinearGAM                                                                                                 
=============================================== ==========================================================
Distribution:                        NormalDist Effective DoF:                                     11.8135
Link Function:                     IdentityLink Log Likelihood:                                  -952.2409
Number of Samples:                          133 AIC:                                             1930.1088
                                                AICc:                                            1933.0789
                                                GCV:                                              609.3811
                                                Scale:                                            512.7965
                                                Pseudo R-Squared:                                   0.7984
==========================================================================================================
Feature Function                  Lambda               Rank         EDoF         P > x        Sig. Code   
================================= ==================== ============ ============ ============ ============
s(0)                              [0.2512]             20           11.8         1.11e-16     ***         
==========================================================================================================
Significance codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

WARNING: Fitting splines and a linear function to a feature introduces a model identifiability problem
         which can cause p-values to appear significant when they are not.

WARNING: p-values calculated in this manner behave correctly for un-penalized models or models with
         known smoothing parameters, but when smoothing parameters have been estimated, the p-values
         are typically lower than they should be, meaning that the tests reject the null too readily.

Fortunately, this routine can be performed directly and results in the following differences between the naive and optimised model.

Plot the GAM fits

fig, ax = plt.subplots(figsize=(10, 6))
plot_fit(gam_raw, X, y, ax, "Unoptimised Fit", c1='orange', c2='green')
plot_fit(gam, X, y, ax, "Optimised Fit")

This is all well and good. We’ve seen an approach to modelling that can capture eccentric patterns in raw data. But how does it work and why should we care?

If you’re familiar with the nomenclature of machine learning, you should think of spline modelling as a variety of feature creation. It generates “synthetic” features over the range of the observed variable. These synthetic features are the splines in question.

Digression on the usage of “Splines”

The history of the term “spline” is related to the history of draftmanship. Historically splines were thin strips of flexible wood or plastic that could be bent or shaped around a weight or “knot” points to express a traceable curve over the space of a “numberline”. The elastic nature of the spline material allowed it to be bent around the knot points of curvature expressing a smooth or continuous bend.

The mathematical technique apes these properties by defining a curve over in an analogous way. We specify “knots” to carve up the support of the random variable \(X\) into portions that require different weighting schemes to represent the outcome \(y\)

spline

Extracting the Splines

We can extract the spline features used in the PyGAM by invoking the following commands to first identify the knot points and create the b-spline basis appropriate for the variable \(X\).

knot_edges=utils.gen_edge_knots(X,dtype='numerical')
knots=np.linspace(knot_edges[0],knot_edges[-1],len(gam.coef_))

splines = utils.b_spline_basis(X, edge_knots=knot_edges, sparse=False)

splines_df = pd.DataFrame(splines, columns=[f'basis_{i}' for i in range(len(gam.coef_))])

splines_df.head(10)
basis_0 basis_1 basis_2 basis_3 basis_4 basis_5 basis_6 basis_7 basis_8 basis_9 basis_10 basis_11 basis_12 basis_13 basis_14 basis_15 basis_16 basis_17 basis_18 basis_19
0 0.166667 0.666667 0.166667 0.000000 0.000000 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
1 0.132997 0.661606 0.205334 0.000063 0.000000 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
2 0.059688 0.594827 0.341426 0.004059 0.000000 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
3 0.030095 0.518726 0.437481 0.013698 0.000000 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
4 0.012374 0.428013 0.527144 0.032470 0.000000 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
5 0.000000 0.040337 0.551431 0.399315 0.008917 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
6 0.000000 0.018232 0.465467 0.492630 0.023671 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
7 0.000000 0.011137 0.418489 0.535407 0.034967 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
8 0.000000 0.000014 0.189310 0.664817 0.145859 0.000000 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
9 0.000000 0.000000 0.120914 0.656897 0.222015 0.000174 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

These spline features range the extent of the of covariate space \(X\) defining “partitions” of the space. The model “learns” to capture the shape of the outcome variable \(y\) by figuring out how to weight the different portion of this spline basis matrix i.e. the linear combination of this basis matrix with the derived coefficients is a model of our outcome variable.

Note how each row is 0 everywhere except within the columns that represent a partition of \(X\). Additionally each row sum to unity. These properties are important because they ensure that any weighted combination of this basis represents the outcome variable in a controlled and quite granular manner. The more splines we use the more control we have of the representation.

Plotting the Weighted Spline

ax = splines_df.dot(gam.coef_).plot(title='Weighted Splines', label='Weighted Combination of Spline Basis', figsize=(10, 6))
ax.set_ylabel("Acceleration")
ax.set_xlabel("Time Steps")
ax.legend();

In this manner we can see how the specification of a spline basis can help us model eccentric curves in an outcome space.

Next we’ll see how to more directly work with the specification of basis splines, passing these feature matrices into Bambi models.

Bayesian Splines with bambi

Under the hood Bambi makes use of the patsy package formula syntax to specify spline basis terms.

knots_6 = np.linspace(0, np.max(X), 6+2)[1:-1]
knots_10 = np.linspace(0, np.max(X), 10+2)[1:-1]
knots_15 = np.linspace(0, np.max(X), 15+2)[1:-1]

df = pd.DataFrame({'X': X.flatten(), 'y': y})
formula1 = 'bs(X, degree=0, knots=knots_6)'
formula2 = 'bs(X, degree=1, knots=knots_6, intercept=True)'
formula3 = 'bs(X, degree=3, knots=knots_6, intercept=True)'
formula4 = 'bs(X, degree=3, knots=knots_10, intercept=True)'
formula5 = 'bs(X, degree=3, knots=knots_15, intercept=True)'
model_spline1 = bmb.Model(f"y ~ {formula1}", df)
model_spline2 = bmb.Model(f"y ~ {formula2}", df)
model_spline3 = bmb.Model(f"y ~ {formula3}", df)
model_spline4 = bmb.Model(f"y ~ {formula4}", df)
model_spline5 = bmb.Model(f"y ~ {formula5}", df)
model_spline5
       Formula: y ~ bs(X, degree=3, knots=knots_15, intercept=True)
        Family: gaussian
          Link: mu = identity
  Observations: 133
        Priors: 
    target = mu
        Common-level effects
            Intercept ~ Normal(mu: -25.5459, sigma: 226.5005)
            bs(X, degree=3, knots=knots_15, intercept=True) ~ Normal(mu: [0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.
                0. 0. 0. 0. 0. 0. 0. 0.], sigma: [1209.5913 1309.9059 1516.6254  914.7421  803.2567
                526.1027  542.4563
              739.4915  590.0538  692.802   786.9096  707.6947  852.6529  792.6311
             1046.3175 1246.4855 1367.9865 1595.1739 1391.2518])
        
        Auxiliary parameters
            sigma ~ HalfStudentT(nu: 4.0, sigma: 48.14)

As you can see here we are specifying a range of different degrees of spline basis. The different degrees corresspond to the smoothness of the overlapping splines. The degree=0 splines mean we specify a piecewise constant basis i.e. 0 or 1 within each region of the partition. But we can add more degrees to see more flexible representations of the space.

model_spline5.build()
model_spline5.graph()

This will become clearer as we plot the various spline basis matrices below.

Plot the Spline Basis

The below functions extract the basis specification from each model and plots the basis design for an increasingly complex series of spline basis matrices.

def plot_spline_basis(basis, X, ax, title="Spline Basis"):
    df = (
        pd.DataFrame(basis)
        .assign(X=X)
        .melt("X", var_name="basis_idx", value_name="y")
    )


    for idx in df.basis_idx.unique():
        d = df[df.basis_idx == idx]
        ax.plot(d["X"], d["y"])
    
    ax.set_title(title)
    return ax

def plot_knots(knots, ax):
    for knot in knots:
        ax.axvline(knot, color="0.1", alpha=0.4)
    return ax



fig, axs = plt.subplots(5, 1, figsize=(9, 20))
axs = axs.flatten()
axs.flatten()
B1 = model_spline1.response_component.design.common[formula1]
plot_spline_basis(B1, df["X"].values, ax=axs[0], title="Piecewise Constant Basis")
plot_knots(knots_6, axs[0]);

B2 = model_spline2.response_component.design.common[formula2]
ax = plot_spline_basis(B2, df["X"].values, axs[1], 
title="Piecewise Linear Basis")
plot_knots(knots_6, axs[1]);

B3 = model_spline3.response_component.design.common[formula3]
ax = plot_spline_basis(B3, df["X"].values, axs[2], 
title="Cubic Spline Basis (6 Knots)")
plot_knots(knots_6, axs[2]);

B4 = model_spline4.response_component.design.common[formula4]
ax = plot_spline_basis(B4, df["X"].values, axs[3], 
title="Cubic Spline Basis (10 Knots)")
plot_knots(knots_10, axs[3]);


B5 = model_spline5.response_component.design.common[formula5]
ax = plot_spline_basis(B5, df["X"].values, axs[4], 
title="Cubic Spline Basis (15 Knots)")
plot_knots(knots_15, axs[4]);

Now that we’ve seen the nature of the modelling splines we’ll fit each model to the data and plot how the weighted splines matrices are able to represent the raw data.

Fit the Individual Spline Models

idata_spline1 = model_spline1.fit(random_seed=random_seed, idata_kwargs={"log_likelihood": True})

idata_spline2 = model_spline2.fit(random_seed=random_seed, idata_kwargs={"log_likelihood": True})

idata_spline3 = model_spline3.fit(random_seed=random_seed, idata_kwargs={"log_likelihood": True})

idata_spline4 = model_spline4.fit(random_seed=random_seed, idata_kwargs={"log_likelihood": True})

idata_spline5 = model_spline5.fit(random_seed=random_seed, idata_kwargs={"log_likelihood": True})

Plotting the fits

Plot the Weighted Mean

def plot_weighted_splines(B, idata, formula, ax, knots):
    posterior_stacked = az.extract(idata)
    wp = posterior_stacked[formula].mean("sample").values

    plot_spline_basis(B * wp.T, df["X"].values, ax)
    ax.plot(df.X.values, np.dot(B, wp.T), color="black", lw=3, label='Weighted Splines')
    plot_knots(knots, ax);
    ax.legend()



fig, axs = plt.subplots(5, 1, figsize=(10, 20))
axs = axs.flatten()
axs.flatten()

plot_weighted_splines(B1, idata_spline1, formula1, axs[0], knots_6)
plot_weighted_splines(B2, idata_spline2, formula2, axs[1], knots_6)
plot_weighted_splines(B3, idata_spline3, formula3, axs[2], knots_6)
plot_weighted_splines(B4, idata_spline4, formula4, axs[3], knots_10)
plot_weighted_splines(B5, idata_spline5, formula5, axs[4], knots_15)

Here we can see how the models with increasingly complex splines are more exactly able to fit the herky jerky trajectory of the outcome variable in each interval. The fewer the intervals, the less flexibility available to the model.

Compare Model Fits

As before we can evaluate these model fits and compare them based on leave-one-out cross validation scores and information theoretic complexity measures.

fig, axs = plt.subplots(2, 1, figsize=(9, 16))
axs = axs.flatten()
models_dict = {"piecewise_constant": idata_spline1, "piecewise_linear": idata_spline2, "cubic_bspline": idata_spline3, "cubic_bspline_10": idata_spline4, 
"cubic_bspline_15": idata_spline5}
df_compare = az.compare(models_dict)
az.plot_compare(df_compare, ax=axs[0])
az.plot_compare(az.compare(models_dict, 'waic'), ax=axs[1])

df_compare
rank elpd_loo p_loo elpd_diff weight se dse warning scale
cubic_bspline_10 0 -612.274608 11.462200 0.000000 9.030402e-01 9.689854 0.000000 True log
cubic_bspline_15 1 -616.252712 15.566839 3.978104 2.408230e-14 9.833120 1.374420 True log
cubic_bspline 2 -634.729956 8.698337 22.455348 5.894750e-03 8.897139 6.637243 True log
piecewise_constant 3 -643.781042 6.981098 31.506433 5.164397e-02 9.770740 8.719803 False log
piecewise_linear 4 -647.267410 6.170543 34.992802 3.942108e-02 7.902853 8.094122 False log

Here we see that the extra complexity of using 15 splines leads to slightly worse performance measures than the less complex but seemingly adequate 10 splines.

new_data = pd.DataFrame({"X": np.linspace(df.X.min(), df.X.max(), num=500)})
    
model_spline4.predict(idata_spline4, data=new_data, 
kind='pps', inplace=True)

idata_spline4
arviz.InferenceData
    • <xarray.Dataset>
      Dimensions:                                              (chain: 4, draw: 1000,
                                                                bs(X, degree=3, knots=knots_10, intercept=True)_dim: 14,
                                                                y_obs: 500)
      Coordinates:
        * chain                                                (chain) int64 0 1 2 3
        * draw                                                 (draw) int64 0 ... 999
        * bs(X, degree=3, knots=knots_10, intercept=True)_dim  (bs(X, degree=3, knots=knots_10, intercept=True)_dim) int64 ...
        * y_obs                                                (y_obs) int64 0 ... 499
      Data variables:
          Intercept                                            (chain, draw) float64 ...
          bs(X, degree=3, knots=knots_10, intercept=True)      (chain, draw, bs(X, degree=3, knots=knots_10, intercept=True)_dim) float64 ...
          y_sigma                                              (chain, draw) float64 ...
          y_mean                                               (chain, draw, y_obs) float64 ...
      Attributes:
          created_at:                  2024-04-06T22:01:45.491675
          arviz_version:               0.17.0
          inference_library:           pymc
          inference_library_version:   5.10.3
          sampling_time:               7.816289663314819
          tuning_steps:                1000
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

    • <xarray.Dataset>
      Dimensions:  (chain: 4, draw: 1000, y_obs: 500)
      Coordinates:
        * chain    (chain) int64 0 1 2 3
        * draw     (draw) int64 0 1 2 3 4 5 6 7 8 ... 992 993 994 995 996 997 998 999
        * y_obs    (y_obs) int64 0 1 2 3 4 5 6 7 8 ... 492 493 494 495 496 497 498 499
      Data variables:
          y        (chain, draw, y_obs) float64 24.86 -16.45 34.86 ... -21.51 -48.83
      Attributes:
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

    • <xarray.Dataset>
      Dimensions:  (chain: 4, draw: 1000, y_obs: 133)
      Coordinates:
        * chain    (chain) int64 0 1 2 3
        * draw     (draw) int64 0 1 2 3 4 5 6 7 8 ... 992 993 994 995 996 997 998 999
        * y_obs    (y_obs) int64 0 1 2 3 4 5 6 7 8 ... 125 126 127 128 129 130 131 132
      Data variables:
          y        (chain, draw, y_obs) float64 -4.14 -4.143 -4.178 ... -4.008 -4.024
      Attributes:
          created_at:                  2024-04-06T22:01:45.597101
          arviz_version:               0.17.0
          inference_library:           pymc
          inference_library_version:   5.10.3
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

    • <xarray.Dataset>
      Dimensions:                (chain: 4, draw: 1000)
      Coordinates:
        * chain                  (chain) int64 0 1 2 3
        * draw                   (draw) int64 0 1 2 3 4 5 ... 994 995 996 997 998 999
      Data variables: (12/17)
          index_in_trajectory    (chain, draw) int64 -11 -49 18 4 23 ... -6 -10 57 73
          energy                 (chain, draw) float64 728.2 728.6 ... 729.9 729.2
          smallest_eigval        (chain, draw) float64 nan nan nan nan ... nan nan nan
          n_steps                (chain, draw) float64 47.0 119.0 55.0 ... 127.0 191.0
          largest_eigval         (chain, draw) float64 nan nan nan nan ... nan nan nan
          step_size_bar          (chain, draw) float64 0.04864 0.04864 ... 0.05827
          ...                     ...
          perf_counter_diff      (chain, draw) float64 0.002146 0.005251 ... 0.008523
          reached_max_treedepth  (chain, draw) bool False False False ... False False
          lp                     (chain, draw) float64 -719.5 -721.5 ... -722.1 -722.2
          process_time_diff      (chain, draw) float64 0.002146 0.005251 ... 0.008496
          diverging              (chain, draw) bool False False False ... False False
          tree_depth             (chain, draw) int64 6 7 6 4 6 6 7 5 ... 5 5 3 4 5 7 8
      Attributes:
          created_at:                  2024-04-06T22:01:45.498837
          arviz_version:               0.17.0
          inference_library:           pymc
          inference_library_version:   5.10.3
          sampling_time:               7.816289663314819
          tuning_steps:                1000
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

    • <xarray.Dataset>
      Dimensions:  (y_obs: 133)
      Coordinates:
        * y_obs    (y_obs) int64 0 1 2 3 4 5 6 7 8 ... 125 126 127 128 129 130 131 132
      Data variables:
          y        (y_obs) float64 0.0 -1.3 -2.7 0.0 -2.7 ... -2.7 10.7 -2.7 10.7
      Attributes:
          created_at:                  2024-04-06T22:01:45.501264
          arviz_version:               0.17.0
          inference_library:           pymc
          inference_library_version:   5.10.3
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

Next we plot the posterior predictive distribution of our observed variable and compare against the observed data. Additionally we plot the 89th and 50% HDI.

ax = az.plot_hdi(new_data['X'], idata_spline4['posterior_predictive']['y'], fill_kwargs={'alpha': 0.2, 'color':'firebrick'}, hdi_prob=0.89, figsize=(10, 8))

az.plot_hdi(new_data['X'], idata_spline4['posterior_predictive']['y'], fill_kwargs={'alpha': 0.8, 'color':'firebrick'}, hdi_prob=0.5)

y_mean = idata_spline4['posterior_predictive']['y'].mean(dim=('chain', 'draw'))

ax.plot(new_data['X'], y_mean, label='Expected posterior predictive', color='k')
ax.set_xlabel("Time Point")
ax.set_ylabel("Acceleration")

ax.scatter(df['X'], df['y'], label='Observed Datapoints')

ax.legend()

ax.set_title("Posterior Predictive Distribution \n Based on 10 Knots");

This represents a good a clean model fit to the observed data using univariate spline smoothers. Next we’ll see another alternative approach to model this outcome variable using approximate gaussian processes.

Gaussian processes

The topic of gaussian processes is rich and detailed. Too rich to be fairly covered in this blog post, so we’ll just say that we’re using a method designed for function approximation that makes use of drawing samples from a multivariate normal distribution under a range of different covariance relationships.

These relationships can be somewhat intuitively interrogated by defining different combinations of covariance relationships with priors over the parameters governing the covariance of a sequence of points. For example consider the following parameterisations.

lengthscale = 3
sigma = 13
cov = sigma**2 * pm.gp.cov.ExpQuad(1, lengthscale)

X = np.linspace(0, 60, 200)[:, None]
K = cov(X).eval()

fig, ax = plt.subplots(figsize=(9, 7))
ax.plot(
    X,
    pm.draw(
        pm.MvNormal.dist(mu=np.zeros(len(K)), cov=K, shape=K.shape[0]), draws=10, random_seed=random_seed
    ).T,
)
plt.title(f"Samples from the GP prior \n lengthscale: {3}, sigma: {13}")
plt.ylabel("y")
plt.xlabel("X");

We’ve specified the range of X to reflect the support of the acceleration example and allowed the draws to be informed by a covariance function we have parameterised using the Exponentiated Quadratic kernel:

\[k(x, x') = \mathrm{exp}\left[ -\frac{(x - x')^2}{2 \ell^2} \right]\]

The patterns exhibited show a good range of “wiggliness” that they should be flexible enough to capture the shape of the acceleration, if we can calibrate the posterior of parameters against the observed data.

Priors on Gaussian Priors

Consider the following specification:

fig, ax = plt.subplots(figsize=(9, 7))
ax.hist(pm.draw(pm.InverseGamma.dist(mu=1, sigma=1), 1000), ec='black', bins=30);

fig, ax = plt.subplots(figsize=(9, 7))
ax.hist(pm.draw(pm.Exponential.dist(lam=1), 1000), ec='black', bins=30);

We use these to specify priors on the Hilbert space approximation of gaussian priors available in the Bambi package.

prior_hsgp = {
    "sigma": bmb.Prior("Exponential", lam=1), # amplitude
    "ell": bmb.Prior("InverseGamma", mu=1, sigma=1) # lengthscale
}

# This is the dictionary we pass to Bambi
priors = {
    "hsgp(X, m=10, c=1)": prior_hsgp,
    "sigma": bmb.Prior("HalfNormal", sigma=4)
}
model_hsgp = bmb.Model("y ~ 0 + hsgp(X, m=10, c=1)", df, priors=priors)
model_hsgp
       Formula: y ~ 0 + hsgp(X, m=10, c=1)
        Family: gaussian
          Link: mu = identity
  Observations: 133
        Priors: 
    target = mu
        HSGP contributions
            hsgp(X, m=10, c=1)
                cov: ExpQuad
                sigma ~ Exponential(lam: 1.0)
                ell ~ InverseGamma(mu: 1.0, sigma: 1.0)
        
        Auxiliary parameters
            sigma ~ HalfNormal(sigma: 4.0)

Here we’ve set the m=10 to determine the number of basis vectors used in the Hilbert space approximation. The idea differs in detail from the spline based approximations we’ve seen, but it’s perhaps useful to think of the process in the same vein.

idata = model_hsgp.fit(inference_method="nuts_numpyro",target_accept=0.95, random_seed=121195, 
idata_kwargs={"log_likelihood": True})
print(idata.sample_stats["diverging"].sum().to_numpy())
Compiling...
Compilation time = 0:00:01.052932
Sampling...
Sampling time = 0:00:01.776702
Transforming variables...
Transformation time = 0:00:00.173566
Computing Log Likelihood...
Log Likelihood time = 0:00:00.147482
0

This model fits and the sampling seems to have worked well.

az.plot_trace(idata, backend_kwargs={"layout": "constrained"}, figsize=(9, 15));

The lengthscale and sigma parameters we have learned by calibrating our priors against the data. The degree to which these parameters are meaningful depend a little on how familar you are with covariance matrix kernels and their properties, so we won’t dwell on the point here.

az.summary(idata, var_names=['hsgp(X, m=10, c=1)_ell', 'hsgp(X, m=10, c=1)_sigma', 'y_sigma', 'hsgp(X, m=10, c=1)_weights'])
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
hsgp(X, m=10, c=1)_ell 3.301 0.666 2.099 4.567 0.016 0.011 1763.0 1815.0 1.0
hsgp(X, m=10, c=1)_sigma 23.675 2.679 18.925 28.847 0.069 0.049 1497.0 2167.0 1.0
y_sigma 20.589 1.121 18.590 22.751 0.019 0.014 3440.0 2485.0 1.0
hsgp(X, m=10, c=1)_weights[0] -131.588 13.784 -157.019 -105.189 0.216 0.154 4071.0 3020.0 1.0
hsgp(X, m=10, c=1)_weights[1] -94.391 18.010 -125.921 -58.020 0.363 0.256 2500.0 2876.0 1.0
hsgp(X, m=10, c=1)_weights[2] 106.121 20.717 69.627 147.247 0.458 0.324 2046.0 2563.0 1.0
hsgp(X, m=10, c=1)_weights[3] 149.289 22.154 106.508 188.107 0.521 0.368 1807.0 2467.0 1.0
hsgp(X, m=10, c=1)_weights[4] -83.399 22.728 -124.812 -40.406 0.539 0.383 1776.0 2517.0 1.0
hsgp(X, m=10, c=1)_weights[5] -125.322 23.273 -169.346 -82.187 0.531 0.377 1917.0 2940.0 1.0
hsgp(X, m=10, c=1)_weights[6] 37.835 21.153 -1.712 77.574 0.512 0.362 1713.0 2577.0 1.0
hsgp(X, m=10, c=1)_weights[7] 94.574 20.745 53.167 132.182 0.423 0.299 2393.0 3105.0 1.0
hsgp(X, m=10, c=1)_weights[8] -1.494 17.184 -31.835 33.063 0.361 0.255 2264.0 2853.0 1.0
hsgp(X, m=10, c=1)_weights[9] -45.442 15.955 -76.649 -16.636 0.289 0.208 3041.0 3183.0 1.0

But again we can sample from the posterior predictive distribution of the outcome variable

model_hsgp.predict(idata, data=new_data, 
kind='pps', inplace=True)

idata
arviz.InferenceData
    • <xarray.Dataset>
      Dimensions:                         (chain: 4, draw: 1000,
                                           hsgp(X, m=10, c=1)_weights_dim: 10,
                                           y_obs: 500)
      Coordinates:
        * chain                           (chain) int64 0 1 2 3
        * draw                            (draw) int64 0 1 2 3 4 ... 996 997 998 999
        * hsgp(X, m=10, c=1)_weights_dim  (hsgp(X, m=10, c=1)_weights_dim) int64 0 ...
        * y_obs                           (y_obs) int64 0 1 2 3 4 ... 496 497 498 499
      Data variables:
          hsgp(X, m=10, c=1)_weights_raw  (chain, draw, hsgp(X, m=10, c=1)_weights_dim) float64 ...
          y_sigma                         (chain, draw) float64 20.86 19.86 ... 21.17
          hsgp(X, m=10, c=1)_sigma        (chain, draw) float64 26.85 24.0 ... 18.06
          hsgp(X, m=10, c=1)_ell          (chain, draw) float64 3.861 3.509 ... 3.76
          hsgp(X, m=10, c=1)_weights      (chain, draw, hsgp(X, m=10, c=1)_weights_dim) float64 ...
          y_mean                          (chain, draw, y_obs) float64 -0.1748 ... ...
          hsgp(X, m=10, c=1)              (chain, draw, y_obs) float64 -0.1748 ... ...
      Attributes:
          created_at:                  2024-04-06T22:02:06.655718
          arviz_version:               0.17.0
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

    • <xarray.Dataset>
      Dimensions:  (chain: 4, draw: 1000, y_obs: 500)
      Coordinates:
        * chain    (chain) int64 0 1 2 3
        * draw     (draw) int64 0 1 2 3 4 5 6 7 8 ... 992 993 994 995 996 997 998 999
        * y_obs    (y_obs) int64 0 1 2 3 4 5 6 7 8 ... 492 493 494 495 496 497 498 499
      Data variables:
          y        (chain, draw, y_obs) float64 -3.183 -11.12 -34.09 ... -9.658 -12.72
      Attributes:
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

    • <xarray.Dataset>
      Dimensions:  (chain: 4, draw: 1000, y_obs: 133)
      Coordinates:
        * chain    (chain) int64 0 1 2 3
        * draw     (draw) int64 0 1 2 3 4 5 6 7 8 ... 992 993 994 995 996 997 998 999
        * y_obs    (y_obs) int64 0 1 2 3 4 5 6 7 8 ... 125 126 127 128 129 130 131 132
      Data variables:
          y        (chain, draw, y_obs) float64 -3.957 -3.957 -3.957 ... -4.013 -4.099
      Attributes:
          created_at:                  2024-04-06T22:02:06.659303
          arviz_version:               0.17.0
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

    • <xarray.Dataset>
      Dimensions:          (chain: 4, draw: 1000)
      Coordinates:
        * chain            (chain) int64 0 1 2 3
        * draw             (draw) int64 0 1 2 3 4 5 6 ... 993 994 995 996 997 998 999
      Data variables:
          acceptance_rate  (chain, draw) float64 0.9484 0.989 0.9772 ... 0.9905 0.9283
          step_size        (chain, draw) float64 0.1631 0.1631 ... 0.1827 0.1827
          diverging        (chain, draw) bool False False False ... False False False
          energy           (chain, draw) float64 671.4 665.4 671.0 ... 670.8 674.0
          n_steps          (chain, draw) int64 31 31 31 15 15 31 ... 15 31 31 15 15 31
          tree_depth       (chain, draw) int64 5 5 5 4 4 5 5 4 4 ... 4 5 5 4 5 5 4 4 5
          lp               (chain, draw) float64 661.7 661.3 665.7 ... 665.6 668.8
      Attributes:
          created_at:                  2024-04-06T22:02:06.658295
          arviz_version:               0.17.0
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

    • <xarray.Dataset>
      Dimensions:  (y_obs: 133)
      Coordinates:
        * y_obs    (y_obs) int64 0 1 2 3 4 5 6 7 8 ... 125 126 127 128 129 130 131 132
      Data variables:
          y        (y_obs) float64 0.0 -1.3 -2.7 0.0 -2.7 ... -2.7 10.7 -2.7 10.7
      Attributes:
          created_at:                  2024-04-06T22:02:06.659574
          arviz_version:               0.17.0
          inference_library:           numpyro
          inference_library_version:   0.13.2
          sampling_time:               1.776702
          modeling_interface:          bambi
          modeling_interface_version:  0.13.0

and plot the model fit to see if it can recover the observed data.

ax = az.plot_hdi(new_data['X'], idata['posterior_predictive']['y'], fill_kwargs={'alpha': 0.2, 'color':'firebrick'}, figsize=(9, 8))

az.plot_hdi(new_data['X'], idata['posterior_predictive']['y'], fill_kwargs={'alpha': 0.8, 'color':'firebrick'}, hdi_prob=0.5)

y_mean = idata['posterior_predictive']['y'].mean(dim=('chain', 'draw'))

ax.plot(new_data['X'], y_mean, label='Expected posterior predictive', color='k')

ax.scatter(df['X'], df['y'], label='Observed Datapoints')

ax.legend()

ax.set_title("Posterior Predictive Distribution \n Based on HSGP approximation")
Text(0.5, 1.0, 'Posterior Predictive Distribution \n Based on HSGP approximation')

And we can compare versus the spline models to see that by the aggregate performance measures our HSGP model seems to come out on top.

models_dict = {"piecewise_constant": idata_spline1, "piecewise_linear": idata_spline2, "cubic_bspline": idata_spline3, "cubic_bspline_10": idata_spline4, 
"cubic_bspline_15": idata_spline5, 'hsgp': idata}
df_compare = az.compare(models_dict)
df_compare
rank elpd_loo p_loo elpd_diff weight se dse warning scale
hsgp 0 -608.819770 9.958827 0.000000 0.873213 12.009698 0.000000 False log
cubic_bspline_10 1 -612.278059 11.465651 3.458289 0.000000 9.688433 2.962396 True log
cubic_bspline_15 2 -616.252712 15.566839 7.432943 0.000000 9.833120 3.457579 True log
cubic_bspline 3 -634.729956 8.698337 25.910186 0.000000 8.897139 7.686648 True log
piecewise_constant 4 -643.781042 6.981098 34.961272 0.069506 9.770740 9.490817 False log
piecewise_linear 5 -647.267410 6.170543 38.447641 0.057282 7.902853 9.200728 False log

Recap

So far we’ve seen how we can use splines and gauassian processes to model highly eccentric functional relationships where the function could be approximated with univariate smoothing routine. Next we’ll show how to use hierarchical modelling over spline fits to extract insight into the data generating process over a family of curves. In particular we’ll focus on the development of insurance loss curves.

Insurance Loss Curves: Hierarchical Spline Models

We draw on car insurance losses data set discussed in Mick Cooney’s Stan case-study, but we simplify things for ourselves considerably by focusing on one type of loss and ensuring that each year under consideration has equal observations of the accruing losses.

loss_df = pd.read_csv('ppauto_pos.csv')
loss_df = loss_df[(loss_df['GRCODE'].isin([43, 353])) & (loss_df['DevelopmentYear'] < 1998)]

loss_df = loss_df[['GRCODE', 'AccidentYear', 'DevelopmentYear', 'DevelopmentLag', 'EarnedPremDIR_B', 'CumPaidLoss_B']]


loss_df.columns = ['grcode', 'acc_year', 'dev_year', 'dev_lag', 'premium', 'cum_loss']
loss_df['lr'] = loss_df['cum_loss'] / loss_df['premium']
loss_df = loss_df[(loss_df['acc_year'] <= 1992) & (loss_df['dev_lag'] <= 6)].reset_index(drop=True)

loss_df['year_code'] = loss_df['acc_year'].astype(str) + '_' + loss_df['grcode'].astype(str)
loss_df.sort_values(by=['year_code', 'acc_year', 'dev_lag'], inplace=True)
loss_df['standardised_premium'] = (loss_df.assign(mean_premium = np.mean(loss_df['premium']))
.assign(std_premium = np.std(loss_df['premium']))
.apply(lambda x: (x['mean_premium'] - x['premium']) /x['std_premium'], axis=1)
)

loss_df.head(12)
grcode acc_year dev_year dev_lag premium cum_loss lr year_code standardised_premium
30 353 1988 1988 1 18793 4339 0.230884 1988_353 -0.347453
31 353 1988 1989 2 18793 9617 0.511733 1988_353 -0.347453
32 353 1988 1990 3 18793 11584 0.616400 1988_353 -0.347453
33 353 1988 1991 4 18793 12001 0.638589 1988_353 -0.347453
34 353 1988 1992 5 18793 12640 0.672591 1988_353 -0.347453
35 353 1988 1993 6 18793 12966 0.689938 1988_353 -0.347453
0 43 1988 1988 1 957 133 0.138976 1988_43 1.722344
1 43 1988 1989 2 957 333 0.347962 1988_43 1.722344
2 43 1988 1990 3 957 431 0.450366 1988_43 1.722344
3 43 1988 1991 4 957 570 0.595611 1988_43 1.722344
4 43 1988 1992 5 957 615 0.642633 1988_43 1.722344
5 43 1988 1993 6 957 615 0.642633 1988_43 1.722344

### Plot the Loss Curves Here we have plotted the developing loss curves from two different coded insurance products.

ax = (loss_df.pivot(index=['dev_lag'], columns=['grcode', 'acc_year'], values='lr')).plot(figsize=(10, 6))
ax.set_title("Loss Ratios by Year");

We want to model these curves collectively as instances of draws from a distribution of loss curves. To do so we will specify a PyMC hierarchical (mixed) spline model. To do so we will have a spline basis for the global hyper parameters beta_g and the individual parameters for each curve. Here we define a convenience function to generate the basis splines.

from patsy import bs, dmatrix

def make_basis_splines(num_knots=3, max_dev=7):
    knot_list = np.linspace(0, max_dev, num_knots+2)[1:-1]
    dev_periods = np.arange(1, max_dev, 1)

    Bi = dmatrix(
        "bs(dev_periods, knots=knots, degree=3, include_intercept=True) - 1",
        {"dev_periods": dev_periods, "knots": knot_list},
    )

    Bg = dmatrix(
        "bs(dev_periods, knots=knots, degree=3, include_intercept=True) - 1",
        {"dev_periods": dev_periods, "knots": knot_list})


    return Bi, Bg

Bi, Bg = make_basis_splines()
Bg
DesignMatrix with shape (6, 7)
  Columns:
    ['bs(dev_periods, knots=knots, degree=3, include_intercept=True)[0]',
     'bs(dev_periods, knots=knots, degree=3, include_intercept=True)[1]',
     'bs(dev_periods, knots=knots, degree=3, include_intercept=True)[2]',
     'bs(dev_periods, knots=knots, degree=3, include_intercept=True)[3]',
     'bs(dev_periods, knots=knots, degree=3, include_intercept=True)[4]',
     'bs(dev_periods, knots=knots, degree=3, include_intercept=True)[5]',
     'bs(dev_periods, knots=knots, degree=3, include_intercept=True)[6]']
  Terms:
    'bs(dev_periods, knots=knots, degree=3, include_intercept=True)' (columns 0:7)
  (to view full data, use np.asarray(this_obj))

Next we specify a model maker function to create the various pooled, unpooled and hierarhical (mixed) models of the insurance curve data. Note that even though we’re specifying a hierarhical model we have not specified a hierarchy over the insurance codes, instead we have added this as a “fixed” effect feature into our regression model.

def make_model(loss_df, num_knots=3, max_dev=7, model_type='mixed'):
    Bi, Bg = make_basis_splines(num_knots, max_dev)
    observed = loss_df['lr'].values
    uniques, unique_codes = pd.factorize(loss_df['year_code'])
    coords= {'years': unique_codes, 
            'splines': list(range(Bi.shape[1])) ,
            'measurement': list(range(6)), 
            'obs': uniques
            }

    with pm.Model(coords=coords) as sp_insur:
        basis_g = pm.MutableData('Bg', np.asfortranarray(Bg))

        tau = pm.HalfCauchy('tau', 1)
        ## Global Hierarchical Spline Terms
        beta_g = pm.Normal("beta_g", mu=0, sigma=tau, 
        dims='splines')
        mu_g = pm.Deterministic("mu_g", pm.math.dot(basis_g, beta_g), dims='measurement')

        ## Individual or Year Specific Spline Modifications
        if model_type in ['mixed', 'unpooled']:
            basis_i = pm.MutableData('Bi', np.asfortranarray(Bi))
            beta = pm.Normal("beta", mu=0, sigma=tau, dims=('splines', 'years'))
            mui = pm.Deterministic("mui", pm.math.dot(basis_i, beta), dims=('measurement', 'years'))
        
        ## Features
        prem = pm.MutableData('prem', loss_df['standardised_premium'].values)
        grcode = pm.MutableData('grcode', loss_df['grcode'] == 43)

        beta_prem = pm.Normal('beta_prem', 0, 1)
        beta_grcode = pm.Normal('beta_grcode', 0, 1)
        mu_prem = beta_prem*prem
        mu_grcode = beta_grcode*grcode

        ## Likelihood
        sigma = pm.TruncatedNormal("sigma", 1, lower=0.05)
        if model_type == 'mixed':
            mu = pm.Deterministic('mu',  mu_grcode + mu_prem + (mu_g.T + mui.T).ravel(), dims='obs')
            lr_likelihood = pm.Normal("lr", mu, sigma, observed=observed, dims=('obs'))
        elif model_type == 'pooled': 
             lr_likelihood = pm.Normal("lr",  np.repeat(mu_g, len(unique_codes)), sigma, observed=observed, dims='obs')
        elif model_type == 'unpooled':
            lr_likelihood = pm.Normal("lr",  mui.T.ravel(), sigma, observed=observed, dims=('obs'))


        ## Sampling
        idata_sp_insur = pm.sample(2000, return_inferencedata=True, target_accept=.99,
        idata_kwargs={"log_likelihood": True})
        idata_sp_insur = pm.sample_posterior_predictive(
            idata_sp_insur,extend_inferencedata=True)

    return idata_sp_insur, sp_insur


idata_sp_insur_unpooled, sp_insur_unpooled = make_model(loss_df, model_type='unpooled')
idata_sp_insur_pooled, sp_insur_pooled = make_model(loss_df, model_type='pooled')
idata_sp_insur_mixed, sp_insur_mixed = make_model(loss_df, model_type='mixed')

The model structure can be seen more clearly in this graph

pm.model_to_graphviz(sp_insur_mixed)

We can extract the effect of the differences grcodes and examine the baseline and annual spline related coefficients.

summary = az.summary(idata_sp_insur_mixed, var_names=['beta_grcode', 'beta_prem', 'beta_g', 'beta'])

summary
mean sd hdi_3% hdi_97% mcse_mean mcse_sd ess_bulk ess_tail r_hat
beta_grcode 0.332 0.073 0.195 0.463 0.001 0.001 4011.0 4925.0 1.0
beta_prem -0.055 0.038 -0.126 0.016 0.001 0.000 3909.0 4486.0 1.0
beta_g[0] 0.075 0.087 -0.094 0.236 0.002 0.001 3029.0 4686.0 1.0
beta_g[1] 0.055 0.177 -0.294 0.370 0.002 0.002 7687.0 6162.0 1.0
beta_g[2] 0.434 0.136 0.172 0.686 0.002 0.001 6150.0 5381.0 1.0
... ... ... ... ... ... ... ... ... ...
beta[6, 1990_43] 0.549 0.104 0.358 0.741 0.002 0.001 4028.0 5314.0 1.0
beta[6, 1991_353] -0.015 0.101 -0.206 0.168 0.002 0.001 3422.0 4702.0 1.0
beta[6, 1991_43] 0.103 0.105 -0.087 0.310 0.002 0.001 3895.0 5159.0 1.0
beta[6, 1992_353] -0.030 0.101 -0.216 0.158 0.002 0.001 3648.0 4773.0 1.0
beta[6, 1992_43] -0.071 0.127 -0.310 0.165 0.002 0.001 3750.0 4700.0 1.0

79 rows × 9 columns

Again we can compare the performance metrics of the various models.

compare_df = az.compare({'unpooled': idata_sp_insur_unpooled, 
            'pooled': idata_sp_insur_pooled, 
            'mixed': idata_sp_insur_mixed})

compare_df
rank elpd_loo p_loo elpd_diff weight se dse warning scale
mixed 0 64.875625 33.749150 0.000000 1.000000e+00 2.180200 0.000000 True log
unpooled 1 41.342083 45.690003 23.533542 7.174095e-12 1.469941 1.377048 True log
pooled 2 -9.279494 5.983067 74.155119 0.000000e+00 5.415796 4.611111 False log

Plot the Posterior Predictive Checks

We can check how well the model can recapture the observed data.

def plot_ppc_splines(idata):
    fig, axs = plt.subplots(5, 2, figsize=(9, 30), sharey=True)
    axs = axs.flatten()
    dev_periods = np.arange(1, 7, 1)
    uniques, unique_codes = pd.factorize(loss_df['year_code'])
    for y, c in zip(unique_codes, range(10)):
        az.plot_hdi(dev_periods, idata['posterior_predictive']['lr'].sel(obs=c), color='firebrick', ax=axs[c], fill_kwargs={'alpha': 0.2}, hdi_prob=.89)
        az.plot_hdi(dev_periods, idata['posterior_predictive']['lr'].sel(obs=c), color='firebrick', ax=axs[c], hdi_prob=0.5)
        axs[c].scatter(dev_periods, loss_df[(loss_df['year_code'] == y)]['lr'], color='k', label='Actual Loss Ratio')
        axs[c].plot(dev_periods, loss_df[(loss_df['year_code'] == y)]['lr'], color='k')
        axs[c].set_title(f"Posterior Predictive Loss Ratio \n 89% and 50% HDI: {y}")
        axs[c].legend()
        axs[c].set_xlabel('Default Lag');

plot_ppc_splines(idata_sp_insur_mixed)

Plot the Hierarchical Components

In the following plot we show similarly how to recapture the observed data, but additionally we decompose the structure of the model in each case and extract baseline forecasts which would be our guide to future loss-ratio development curves in lieu of any other information.

mu_g = idata_sp_insur_mixed.posterior.stack(draws=("chain", "draw"))["mu_g"]

mu_i = idata_sp_insur_mixed.posterior.stack(draws=("chain", "draw"))["mui"]

mu  = idata_sp_insur_mixed.posterior.stack(draws=("chain", "draw"))['mu']

beta_grcode = idata_sp_insur_mixed.posterior.stack(draws=("chain", "draw"))['beta_grcode']

dev_periods = np.arange(1, 7, 1)
uniques, unique_codes = pd.factorize(loss_df['year_code'])

mosaic = """
         AB
         CD
         EF
         GH
         IJ
         KK
"""
fig, axs = plt.subplot_mosaic(mosaic, sharey=True, 
figsize=(9, 30))
axs = [axs[k] for k in axs.keys()] 

mu_g_mean = mu_g.mean(dim='draws')
for y, c in zip(unique_codes, range(10)):
    group_effect = 0
    if '43' in y: 
        group_effect = beta_grcode.mean().item()
    mu_i_mean = mu_i.sel(years=y).mean(dim='draws')
    axs[c].plot(dev_periods, group_effect + mu_g_mean.values + mu_i_mean.values, label='Combined + E(grp_effect)', color='purple', linewidth=3.5)
    axs[c].plot(dev_periods, group_effect + mu_g_mean.values, label='E(Hierarchical Baseline)', color='red', linestyle='--')
    axs[c].plot(dev_periods,  group_effect + mu_i_mean.values, label='E(Year Specific Adjustment term)', color='blue', linestyle='--')
    axs[c].scatter(dev_periods, loss_df[(loss_df['year_code'] == y)]['lr'], color='k', label='Actual Loss Ratio')
    axs[c].legend()
    az.plot_hdi(dev_periods,mu.sel(obs=c).T  , ax=axs[c], color='firebrick', fill_kwargs={'alpha': 0.2})
    az.plot_hdi(dev_periods, mu.sel(obs=c).T , ax=axs[c], color='firebrick', fill_kwargs={'alpha': 0.5}, hdi_prob=.50)
    axs[c].set_title(f"Components for Year {y}")

axs[10].plot(dev_periods, mu_g_mean.values, label='E(Hierarchical Baseline)', color='black')
axs[10].plot(dev_periods, mu_g_mean.values + group_effect, label='E(Hierarchical Baseline) + E(grp_effect)', color='black', linestyle='--')
az.plot_hdi(dev_periods, mu_g.T.values, color='slateblue', ax=axs[10], fill_kwargs={'alpha': 0.2})
az.plot_hdi(dev_periods, mu_g.T.values + group_effect, color='magenta', ax=axs[10], fill_kwargs={'alpha': 0.2})
az.plot_hdi(dev_periods, mu_g.T.values, color='slateblue', ax=axs[10], hdi_prob=.5)
az.plot_hdi(dev_periods, mu_g.T.values  + group_effect, color='magenta', ax=axs[10], hdi_prob=.5)
axs[10].set_title("Baseline Forecast Loss Ratio \n with Uncertainty Intervals")
axs[10].legend();

This plot is a bit clunky, because we’re mixing expectations and posterior distributions over the parameters. The point is just to highlight the “compositional” structure of our model. A better way to interrogate the implications of the model is to “push” forward different data through the posterior predictive distribution and derive a kind of ceteris paribus rule for accrual of losses. We d

with sp_insur_mixed: 
    pm.set_data({'grcode': np.ones(len(loss_df)), 
    })
    idata_43 = pm.sample_posterior_predictive(idata_sp_insur_mixed, var_names=['lr'], extend_inferencedata =True)

with sp_insur_mixed: 
    pm.set_data({'grcode': np.zeros(len(loss_df))})
    idata_353 = pm.sample_posterior_predictive(idata_sp_insur_mixed, var_names=['lr'], extend_inferencedata=True)

idata_353
Sampling: [lr]
100.00% [8000/8000 00:00<00:00]
Sampling: [lr]
100.00% [8000/8000 00:00<00:00]
arviz.InferenceData
    • <xarray.Dataset>
      Dimensions:      (chain: 4, draw: 2000, splines: 7, years: 10, measurement: 6,
                        obs: 60)
      Coordinates:
        * chain        (chain) int64 0 1 2 3
        * draw         (draw) int64 0 1 2 3 4 5 6 ... 1994 1995 1996 1997 1998 1999
        * splines      (splines) int64 0 1 2 3 4 5 6
        * years        (years) <U8 '1988_353' '1988_43' ... '1992_353' '1992_43'
        * measurement  (measurement) int64 0 1 2 3 4 5
        * obs          (obs) int64 0 0 0 0 0 0 1 1 1 1 1 1 ... 8 8 8 8 8 8 9 9 9 9 9 9
      Data variables:
          beta_g       (chain, draw, splines) float64 0.09245 -0.06587 ... 0.4671
          beta         (chain, draw, splines, years) float64 0.06975 ... -0.06849
          beta_prem    (chain, draw) float64 -0.04083 -0.1108 ... 0.02838 -0.09618
          beta_grcode  (chain, draw) float64 0.3686 0.3849 0.3135 ... 0.2748 0.3049
          tau          (chain, draw) float64 0.2592 0.2726 0.28 ... 0.2711 0.2752
          sigma        (chain, draw) float64 0.05167 0.06453 ... 0.0648 0.05801
          mu_g         (chain, draw, measurement) float64 0.09245 0.3975 ... 0.4671
          mui          (chain, draw, measurement, years) float64 0.06975 ... -0.06849
          mu           (chain, draw, obs) float64 0.1764 0.5378 ... 0.9267 0.8547
      Attributes:
          created_at:                 2024-04-06T22:03:49.703712
          arviz_version:              0.17.0
          inference_library:          pymc
          inference_library_version:  5.10.3
          sampling_time:              42.2772331237793
          tuning_steps:               1000

    • <xarray.Dataset>
      Dimensions:  (chain: 4, draw: 2000, obs: 60)
      Coordinates:
        * chain    (chain) int64 0 1 2 3
        * draw     (draw) int64 0 1 2 3 4 5 6 7 ... 1993 1994 1995 1996 1997 1998 1999
        * obs      (obs) int64 0 0 0 0 0 0 1 1 1 1 1 1 2 ... 7 8 8 8 8 8 8 9 9 9 9 9 9
      Data variables:
          lr       (chain, draw, obs) float64 0.1192 0.421 0.6624 ... 0.9543 0.8841
      Attributes:
          created_at:                 2024-04-06T22:03:51.789957
          arviz_version:              0.17.0
          inference_library:          pymc
          inference_library_version:  5.10.3

    • <xarray.Dataset>
      Dimensions:  (chain: 4, draw: 2000, obs: 60)
      Coordinates:
        * chain    (chain) int64 0 1 2 3
        * draw     (draw) int64 0 1 2 3 4 5 6 7 ... 1993 1994 1995 1996 1997 1998 1999
        * obs      (obs) int64 0 0 0 0 0 0 1 1 1 1 1 1 2 ... 7 8 8 8 8 8 8 9 9 9 9 9 9
      Data variables:
          lr       (chain, draw, obs) float64 1.488 1.916 1.887 ... 1.581 1.85 1.205
      Attributes:
          created_at:                 2024-04-06T22:03:49.970357
          arviz_version:              0.17.0
          inference_library:          pymc
          inference_library_version:  5.10.3

    • <xarray.Dataset>
      Dimensions:                (chain: 4, draw: 2000)
      Coordinates:
        * chain                  (chain) int64 0 1 2 3
        * draw                   (draw) int64 0 1 2 3 4 5 ... 1995 1996 1997 1998 1999
      Data variables: (12/17)
          index_in_trajectory    (chain, draw) int64 16 85 59 64 ... -77 143 102 142
          energy                 (chain, draw) float64 -43.01 -28.79 ... -31.02 -38.76
          smallest_eigval        (chain, draw) float64 nan nan nan nan ... nan nan nan
          n_steps                (chain, draw) float64 127.0 127.0 ... 255.0 255.0
          largest_eigval         (chain, draw) float64 nan nan nan nan ... nan nan nan
          step_size_bar          (chain, draw) float64 0.03773 0.03773 ... 0.03628
          ...                     ...
          perf_counter_diff      (chain, draw) float64 0.009788 0.01019 ... 0.01847
          reached_max_treedepth  (chain, draw) bool False False False ... False False
          lp                     (chain, draw) float64 80.47 71.15 ... 73.09 77.75
          process_time_diff      (chain, draw) float64 0.009787 0.01013 ... 0.01846
          diverging              (chain, draw) bool False False False ... False False
          tree_depth             (chain, draw) int64 7 7 8 8 7 7 7 8 ... 7 9 7 7 8 8 8
      Attributes:
          created_at:                 2024-04-06T22:03:49.717087
          arviz_version:              0.17.0
          inference_library:          pymc
          inference_library_version:  5.10.3
          sampling_time:              42.2772331237793
          tuning_steps:               1000

    • <xarray.Dataset>
      Dimensions:  (obs: 60)
      Coordinates:
        * obs      (obs) int64 0 0 0 0 0 0 1 1 1 1 1 1 2 ... 7 8 8 8 8 8 8 9 9 9 9 9 9
      Data variables:
          lr       (obs) float64 0.2309 0.5117 0.6164 0.6386 ... 0.859 0.9038 0.9244
      Attributes:
          created_at:                 2024-04-06T22:03:49.719921
          arviz_version:              0.17.0
          inference_library:          pymc
          inference_library_version:  5.10.3

    • <xarray.Dataset>
      Dimensions:       (Bg_dim_0: 6, Bg_dim_1: 7, Bi_dim_0: 6, Bi_dim_1: 7,
                         prem_dim_0: 60, grcode_dim_0: 60)
      Coordinates:
        * Bg_dim_0      (Bg_dim_0) int64 0 1 2 3 4 5
        * Bg_dim_1      (Bg_dim_1) int64 0 1 2 3 4 5 6
        * Bi_dim_0      (Bi_dim_0) int64 0 1 2 3 4 5
        * Bi_dim_1      (Bi_dim_1) int64 0 1 2 3 4 5 6
        * prem_dim_0    (prem_dim_0) int64 0 1 2 3 4 5 6 7 ... 52 53 54 55 56 57 58 59
        * grcode_dim_0  (grcode_dim_0) int64 0 1 2 3 4 5 6 7 ... 53 54 55 56 57 58 59
      Data variables:
          Bg            (Bg_dim_0, Bg_dim_1) float64 1.0 0.0 0.0 0.0 ... 0.0 0.0 1.0
          Bi            (Bi_dim_0, Bi_dim_1) float64 1.0 0.0 0.0 0.0 ... 0.0 0.0 1.0
          prem          (prem_dim_0) float64 -0.3475 -0.3475 -0.3475 ... -1.572 -1.572
          grcode        (grcode_dim_0) float64 0.0 0.0 0.0 0.0 0.0 ... 1.0 1.0 1.0 1.0
      Attributes:
          created_at:                 2024-04-06T22:03:49.720862
          arviz_version:              0.17.0
          inference_library:          pymc
          inference_library_version:  5.10.3

Even here though we want to average the curves over the specific years in the data and abstract a view of the model implications under different counterfactual settings. Here we define a helper function to effect this step.

def get_posterior_predictive_curve(idata, prem=2, grcode=1):
    weighted_splines = [np.dot(np.asfortranarray(Bi), az.extract(idata['posterior']['beta'])['beta'].values[:, :, i]) for i in range(1000)]

    weighted_splines_1 = [np.dot(np.asfortranarray(Bg), az.extract(idata['posterior']['beta_g'])['beta_g'].values[:, i]) for i in range(1000)]

    beta_grcode = az.extract(idata['posterior']['beta_grcode'])['beta_grcode']

    beta_prem = az.extract(idata['posterior']['beta_prem'])['beta_prem']
    df1 = pd.DataFrame([beta_prem.values[i]*prem + beta_grcode.values[i]*grcode for i in range(1000)]).T

    ## Crucial step averaging over the years to get
    ## a view of the development period
    df = pd.concat([pd.DataFrame(weighted_splines_1[i].T + weighted_splines[i].T).mean() for i in range(1000)], axis=1)

    df = df1.iloc[0] + df

    return df

pred_df_1 = get_posterior_predictive_curve(idata_43, prem=1, grcode=1)

pred_df_0 = get_posterior_predictive_curve(idata_353, prem=1, grcode=0) 

fig, ax = plt.subplots(figsize=(9, 7), sharey=True)


ax.plot(pred_df_1, color='firebrick', alpha=0.1)
ax.plot(pred_df_0, color='slateblue', alpha=0.2);
ax.plot(pred_df_0.mean(axis=1), linestyle='-', color='k', linewidth=4, label='grcode 353 prem 1')

ax.plot(pred_df_1.mean(axis=1), linestyle='--', color='grey', linewidth=4, label='grcode 43 prem 1')

ax.set_title("Posterior Samples of the Trajectories \n Under different Counterfactual settings")
ax.set_ylabel("Loss Ratio")
ax.set_xlabel("Development Period")
ax.legend();

Conclusion

We’ve seen the application of splines as univariate smoothers to approximate wiggly curves of arbitrary shape. We’ve also tried gaussian process approximations of the same univariate functions. The suggested flexibility of both methods is a strength, but we need to be careful where splines have a tendency to over-fit to individual curves. As such we have tried to show that we incorporate spline basis modelling in a hierarchical bayesian model and recover very compelling posterior predictive checks and additionally derive predictiions from the mixed variant of the hierarhical model which helps us understand the implications of the data for generic forecasts.